mtimes, *

Quaternion multiplication

Description

example

quatC = A*B implements quaternion multiplication if either A or B is a quaternion. Either A or B must be a scalar.

You can use quaternion multiplication to compose rotation operators:

• To compose a sequence of frame rotations, multiply the quaternions in the order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the order pq. The rotation operator becomes ${\left(pq\right)}^{\ast }v\left(pq\right)$, where v represents the object to rotate specified in quaternion form. * represents conjugation.

• To compose a sequence of point rotations, multiply the quaternions in the reverse order of the desired sequence of rotations. For example, to apply a p quaternion followed by a q quaternion, multiply in the reverse order, qp. The rotation operator becomes $\left(qp\right)v{\left(qp\right)}^{\ast }$.

Examples

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Create a 4-by-1 column vector, A, and a scalar, b. Multiply A times b.

A = quaternion(randn(4,4))
A=4×1 quaternion array
0.53767 +  0.31877i +   3.5784j +   0.7254k
1.8339 -   1.3077i +   2.7694j - 0.063055k
-2.2588 -  0.43359i -   1.3499j +  0.71474k
0.86217 +  0.34262i +   3.0349j -  0.20497k

b = quaternion(randn(1,4))
b = quaternion
-0.12414 +  1.4897i +   1.409j +  1.4172k

C = A*b
C=4×1 quaternion array
-6.6117 +   4.8105i +  0.94224j -   4.2097k
-2.0925 +   6.9079i +   3.9995j -   3.3614k
1.8155 -   6.2313i -    1.336j -     1.89k
-4.6033 +   5.8317i + 0.047161j -    2.791k

Input Arguments

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Input to multiply, specified as a quaternion, array of quaternions, real scalar, or array of real scalars.

If B is nonscalar, then A must be scalar.

Data Types: quaternion | single | double

Input to multiply, specified as a quaternion, array of quaternions, real scalar, or array of real scalars.

If A is nonscalar, then B must be scalar.

Data Types: quaternion | single | double

Output Arguments

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Quaternion product, returned as a quaternion or array of quaternions.

Data Types: quaternion

Algorithms

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Quaternion Multiplication by a Real Scalar

Given a quaternion

$q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$

the product of q and a real scalar β is

$\beta q=\beta {a}_{\text{q}}+\beta {b}_{\text{q}}\text{i}+\beta {c}_{\text{q}}\text{j}+\beta {d}_{\text{q}}\text{k}$

Quaternion Multiplication by a Quaternion Scalar

The definition of the basis elements for quaternions,

${\text{i}}^{2}={\text{j}}^{2}={\text{k}}^{2}=\text{ijk}=-1\text{\hspace{0.17em}},$

can be expanded to populate a table summarizing quaternion basis element multiplication:

 1 i j k 1 1 i j k i i −1 k −j j j −k −1 i k k j −i −1

When reading the table, the rows are read first, for example: ij = k and ji = −k.

Given two quaternions, $q={a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k,}$ and $p={a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}$, the multiplication can be expanded as:

$\begin{array}{l}z=pq=\left({a}_{\text{p}}+{b}_{\text{p}}\text{i}+{c}_{\text{p}}\text{j}+{d}_{\text{p}}\text{k}\right)\left({a}_{\text{q}}+{b}_{\text{q}}\text{i}+{c}_{\text{q}}\text{j}+{d}_{\text{q}}\text{k}\right)\\ \text{ }\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}+{b}_{\text{p}}{b}_{\text{q}}{\text{i}}^{2}+{b}_{\text{p}}{c}_{\text{q}}\text{ij}+{b}_{\text{p}}{d}_{\text{q}}\text{ik}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}+{c}_{\text{p}}{b}_{\text{q}}\text{ji}+{c}_{\text{p}}{c}_{\text{q}}{\text{j}}^{2}+{c}_{\text{p}}{d}_{\text{q}}\text{jk}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{ki}+{d}_{\text{p}}{c}_{\text{q}}\text{kj}+{d}_{\text{p}}{d}_{\text{q}}{\text{k}}^{2}\end{array}$

You can simplify the equation using the quaternion multiplication table:

$\begin{array}{l}z=pq\text{\hspace{0.17em}}={a}_{\text{p}}{a}_{\text{q}}+{a}_{\text{p}}{b}_{\text{q}}\text{i}+{a}_{\text{p}}{c}_{\text{q}}\text{j}+{a}_{\text{p}}{d}_{\text{q}}\text{k}\\ \text{ }\text{ }+{b}_{\text{p}}{a}_{\text{q}}\text{i}-{b}_{\text{p}}{b}_{\text{q}}+{b}_{\text{p}}{c}_{\text{q}}\text{k}-{b}_{\text{p}}{d}_{\text{q}}\text{j}\\ \text{ }\text{ }+{c}_{\text{p}}{a}_{\text{q}}\text{j}-{c}_{\text{p}}{b}_{\text{q}}\text{k}-{c}_{\text{p}}{c}_{\text{q}}+{c}_{\text{p}}{d}_{\text{q}}\text{i}\\ \text{ }\text{ }+{d}_{\text{p}}{a}_{\text{q}}k+{d}_{\text{p}}{b}_{\text{q}}\text{j}-{d}_{\text{p}}{c}_{\text{q}}\text{i}-{d}_{\text{p}}{d}_{\text{q}}\end{array}$

References

[1] Kuipers, Jack B. Quaternions and Rotation Sequences: A Primer with Applications to Orbits, Aerospace, and Virtual Reality. Princeton, NJ: Princeton University Press, 2007.